Rational points on elliptic curves. (English) Zbl 0752.14034

Silverman, Joseph H.; Tate, John

Undergraduate Texts in Mathematics. New York: Springer-Verlag. x, 281 p. (1992).

The book gives a good introduction for students which are interested in Diophantine equations and arithmetic geometry. It is based on lectures of J. Tate from 1961. It contains a lot of exercises. Often further developments and applications are explained, for instance Lenstra’s algorithm for factorisation of integers using elliptic curves.
The book starts with the geometry and group structure of elliptic curves. It contains the Nagell-Lutz theorem describing points if finite order, the Mordell-Weil theorem on the finite generation of the group of rational points, the Thue-Siegel theorem on the finiteness of the set of integer points. Also points over finite fields are considered. — At the end one finds complex multiplication and Galois representations associated to torsion points.
The algebraic geometry needed for the purpose of the book (for instance Bézout’s theorem) is presented in an appendix.

Reviewer: Gerhard Pfister (Berlin)

MSC:

11G05	Elliptic curves over global fields
11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
14-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
11G15	Complex multiplication and moduli of abelian varieties
11D25	Cubic and quartic Diophantine equations
14G05	Rational points
14G15	Finite ground fields in algebraic geometry
14H52	Elliptic curves

Keywords:

Diophantine equations; elliptic curves; Nagell-Lutz theorem; Mordell-Weil theorem; rational points; Thue-Siegel theorem; integer points; finite fields; complex multiplication; torsion points